Which Set Of Numbers Could Represent The Lengths Of The Sides Of A Right Triangle?A. 7, 24, 25 B. 6, 9, 11 C. 9, 12, 16 D. 10, 15, 20

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Which Set of Numbers Could Represent the Lengths of the Sides of a Right Triangle?

Understanding the Basics of Right Triangles

A right triangle is a triangle with one angle that measures 90 degrees. The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs. In a right triangle, the Pythagorean theorem states that the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the legs (a and b). This can be expressed as: a^2 + b^2 = c^2.

Applying the Pythagorean Theorem

To determine which set of numbers could represent the lengths of the sides of a right triangle, we need to apply the Pythagorean theorem to each option. Let's start by examining each option:

Option A: 7, 24, 25

  • Checking if the numbers satisfy the Pythagorean theorem: We need to check if the square of the longest side (25) is equal to the sum of the squares of the other two sides (7 and 24). Let's calculate: 7^2 + 24^2 = 49 + 576 = 625. Since 25^2 = 625, this set of numbers satisfies the Pythagorean theorem.

Option B: 6, 9, 11

  • Checking if the numbers satisfy the Pythagorean theorem: We need to check if the square of the longest side (11) is equal to the sum of the squares of the other two sides (6 and 9). Let's calculate: 6^2 + 9^2 = 36 + 81 = 117. Since 11^2 = 121, this set of numbers does not satisfy the Pythagorean theorem.

Option C: 9, 12, 16

  • Checking if the numbers satisfy the Pythagorean theorem: We need to check if the square of the longest side (16) is equal to the sum of the squares of the other two sides (9 and 12). Let's calculate: 9^2 + 12^2 = 81 + 144 = 225. Since 16^2 = 256, this set of numbers does not satisfy the Pythagorean theorem.

Option D: 10, 15, 20

  • Checking if the numbers satisfy the Pythagorean theorem: We need to check if the square of the longest side (20) is equal to the sum of the squares of the other two sides (10 and 15). Let's calculate: 10^2 + 15^2 = 100 + 225 = 325. Since 20^2 = 400, this set of numbers does not satisfy the Pythagorean theorem.

Conclusion

Based on the calculations, only option A (7, 24, 25) satisfies the Pythagorean theorem, which means that this set of numbers could represent the lengths of the sides of a right triangle.
Frequently Asked Questions About Right Triangles and the Pythagorean Theorem

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a mathematical concept that describes the relationship between the lengths of the sides of a right triangle. It states that the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the legs (a and b). This can be expressed as: a^2 + b^2 = c^2.

Q: How do I apply the Pythagorean theorem to a right triangle?

A: To apply the Pythagorean theorem, you need to identify the lengths of the sides of the right triangle. Let's say you have a right triangle with legs of length 3 and 4, and a hypotenuse of length 5. You can use the Pythagorean theorem to check if the triangle is a right triangle by calculating: 3^2 + 4^2 = 9 + 16 = 25. Since 5^2 = 25, this triangle is indeed a right triangle.

Q: What are the different types of triangles?

A: There are several types of triangles, including:

  • Acute triangles: Triangles with all angles less than 90 degrees.
  • Right triangles: Triangles with one angle that measures 90 degrees.
  • Obtuse triangles: Triangles with one angle that measures more than 90 degrees.
  • Equilateral triangles: Triangles with all sides of equal length.
  • Isosceles triangles: Triangles with two sides of equal length.

Q: How do I determine if a triangle is a right triangle?

A: To determine if a triangle is a right triangle, you can use the Pythagorean theorem. If the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs, then the triangle is a right triangle.

Q: What are some real-world applications of the Pythagorean theorem?

A: The Pythagorean theorem has many real-world applications, including:

  • Building design: Architects use the Pythagorean theorem to calculate the height of buildings and the length of shadows.
  • Surveying: Surveyors use the Pythagorean theorem to calculate distances and angles between landmarks.
  • Physics: Physicists use the Pythagorean theorem to calculate distances and velocities in problems involving motion.
  • Engineering: Engineers use the Pythagorean theorem to calculate stresses and strains in materials.

Q: Can I use the Pythagorean theorem to calculate the length of a side of a triangle if I know the lengths of the other two sides?

A: Yes, you can use the Pythagorean theorem to calculate the length of a side of a triangle if you know the lengths of the other two sides. For example, if you know the lengths of the legs of a right triangle, you can use the Pythagorean theorem to calculate the length of the hypotenuse.

Q: What are some common mistakes to avoid when using the Pythagorean theorem?

A: Some common mistakes to avoid when using the Pythagorean theorem include:

  • Rounding errors: Make sure to calculate the squares of the lengths of the sides exactly, without rounding.
  • Sign errors**: Make sure to use the correct signs for the lengths of the sides.
  • Incorrect application: Make sure to apply the Pythagorean theorem correctly, using the correct formula and variables.

Conclusion

The Pythagorean theorem is a powerful tool for calculating the lengths of the sides of right triangles. By understanding the theorem and how to apply it, you can solve a wide range of problems in mathematics, science, and engineering.